Courant Institute, New York University, New York, NY 10012, USA

Uri M. Ascher

Department of Computer Science, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

Abstract

The goal of this work is to develop an efficient multigrid solver for the steady-state incompressible Navier-Stokes equations on non-staggered grids. The pressure Poisson equation (PPE) is used instead of the continuity equation in order to avoid odd-even pressure instability. The differential order of the resulting system of equations is higher than that of the original system, so additional boundary conditions are needed. For this, Neumann-type boundary conditions for the pressure can be derived from the given boundary conditions (sufficient for the original formulation) using the momentum and continuity equations.

The main achievements of this work are:

- A method of discretizing the Neumann-type boundary conditions for pressure is developed using a finite volume approach. A relaxation scheme capable of treating efficiently the resulting difference equations is constructed. The speed of convergence of the resulting FMG multigrid solver for low Reynolds numbers is comparable to that for the Poisson equation with Dirichlet boundary conditions.
- A compact, second order accurate, difference scheme approximating the momentum equations for high Reynolds numbers is developed.